On ?properties of rings with a finite number of zero divisors?

Abstract

1. N. GANESAN proved,in [1], that if R is a commutative ring containing n + 1 zero divisors, where n is a positive integer, then the order of R is less than or equal to (n + 1) 2. In this paper, we will prove that if R is an arbitrary ring having n + 1 left (right) zero divisors, then R has at most (n + 1) 2 elements. We will also prove that if 1 e R, the order of R is even less than (n + 1) 2 unless n + 1 is a power of a prime p, and every minimal right ideal I has the property 12 = 0. Actually in case R contains a minimal right ideal I which is not nilpotent, the 1 2 order of R is less than or equal to -~-n + n + 1. Finally we prove that if the set of all left (right) zero divisors of R has the infinite cardinality, say ~t then the cardinality of R is precisely ~.2. An element x of an arbitrary ring R is called a left (right) zero divisor if there is a e R, a 4: 0, such that ax = O(xa = 0). Theorem I. Any ring having only n + 1 left (right) zero divisors, where n is a positive integer, is necessarily finite and does not coniain more than (n + 1) 2 elements.Proof. If a ~ R, let R a be the right annulator of a, i. e. the set of all elements r such that ar = 0. Since R contains n + 1 left zero divisors, there is a e R, a ~ 0, such that the order of R a is at most n + 1. Let y 4 0 be an element of Ra. Then the order of yR is at most n + 1 since yR is a subset of Ra. The order of yR is the order of R/Ry since yR ~ R/Ry. Thus the order of R is the product of the order ofyR and the order of Ry, which is less than or equal to (n + 1) 2. Theorem II. Let R be a ring with 1. If R contains n + 1 left (right) zero divisors then the order of R is even less then (n + 1) 2 unless n + 1 is a power of a prime p, and every minimal right (left) ideal I has the property I 2 = O.Proof. Let I be a minimal right ideal of R such that 12 + 0. Then there is an idempotent 0 ~: e e I such that Rec~eR = 0. The order of R e is less than or equal to n + 1 since every element of Re is a left zero divisor. Similarly every element of eR is also a left zero divisor since 1 -e :~ 0 annihilates eR. Let ml be the order ofR e and m2 be the order of oR. Then ml • m 2 is the order of R and m 1 + m 2 ~< n + 2. Since 2m 1 • m 2 _< m 2 + m 2, 4ml • m 2 ~ (ml + m2) 2, i. e. m t • m2 < = 2 -J < \~-2--/ =-4-+ n + 1. Hence the order of R is less than (n + 1) 2. Now, suppose that if I is a minimal right ideal of R then 12 = 0. If the

On zero-divisor graphs of small finite commutative rings

“…It was proved in [9] that if |Z(R)| = m and |R| = m 2 , then m = p r for some integer r 1 and some prime p. These rings were categorized in [6] by the use of two constructions. When the ring R is commutative with 1, then there are only two such rings (up to isomorphism) for m = p r :…”

Finite rings with many zero divisors

“…We give a description of the non-local rings with identity for which |A| = (n + l) 2 /4 or |A| = n(n + 2)/4, and also of the rings with the condition |A| = (n -k)(n -/), where k e {1,2}, / G {0, 1} (in this case |A| < n 3 / 2 , i.e., n > tf\Rf). Moreover, some results from [3,5,7,8] are generalized.…”

“…If an associative ring R contains n > 1 left (right) zero divisors, including zero, then \R\> n > ^J\R\ (for rings with identity see [1,2], a short proof for an arbitrary ring is given in [3]). Rings with the condition n = ^J\R\ (Corbasian rings) have been described in [4,5] (up to Galois rings) and rings with the conditions \J\R\ 2 > n> \f\ft\, in [6] (up to Corbasian rings).…”

“…Rings with the condition n = ^J\R\ (Corbasian rings) have been described in [4,5] (up to Galois rings) and rings with the conditions \J\R\ 2 > n> \f\ft\, in [6] (up to Corbasian rings). If n = y/\R\> * en n & a prime power [3,5]. For integers 5 and t, 0 < 5 < i, there exists an indecomposable ring R such that |J?| = p i and n = p 5 , where p is prime, if and only if (t -s) \ s [7].…”

Higher commutators, ideals and cardinality

“…A number of papers in the literature relate the cardinality of a ring to that of special subsets. To mention a few of these, in [7], Koh shows that a ring is finite if it has finitely many left zero divisors, and at least two, Hirano extended this to finitely many two-sided zero divisors [3], and recently we considered the relation between the cardinality of a ring and that of the ideal generated by various subsets of nilpotent elements [8]. We have also studied the relation between the cardinality of a ring and that of the subring generated by the set of integral elements or the set of symmetric elements in a ring with involution [9].…”